warm-up:
DESCRIPTION
Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Warm-up:. Evaluate the integrals. 1) 2). Integration by Parts. Section 8.2 Objective: To integrate problems without a u -substitution. Integration by Parts. - PowerPoint PPT PresentationTRANSCRIPT
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
Warm-up:Evaluate the integrals.
1)
2)
dx
xex
73
dxx
x )13
1(
2
Cxex ln73
Cxx
3
sin
3
2 12
3
Integration by Parts
Section 8.2
Objective: To integrate problems without a u-substitution
Integration by Parts• When integrating the product of two functions, we
often use a u-substitution to make the problem easier to integrate. Sometimes this is not possible. We need another way to solve such problems.
)()( xgxf
Integration by Parts• As a first step, we will take the derivative of )()( xgxf
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
)()()()()()( // xfxgxgxfxgxf
Integration by Parts• As a first step, we will take the derivative of
)()()()()()( // xfxgxgxfxgxfdx
d
)()( xgxf
)()()()()()( // xfxgxgxfxgxfdx
d
)()()()()()( // xfxgxgxfxgxf
)()()()()()( // xgxfxfxgxgxf
Integration by Parts• Now lets make some substitutions to make this easier
to apply.)(xgv )(xfu
)()()()()()( // xgxfxfxgxgxf
)(/ xgdv )(/ xfdu
udvvduuv
Integration by Parts• This is the way we will look at these problems.
• The two functions in the original problem we are integrating are u and dv. The first thing we will do is to choose one function for u and the other function will be dv.
)(xgv )(xfu
)(/ xgdv )(/ xfdu udvvduuv
Example 1• Use integration by parts to evaluate xdxx cos
Example 1• Use integration by parts to evaluate
xu xdxdv cos
xdxx cos
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
xdxxxxdxx sinsincos
Example 1• Use integration by parts to evaluate
xv sin
xu xdxdv cos
dxdu
xdxx cos
xdxxxxdxx sinsincos
Cxxxxdxx cossincos
Guidelines
• The first step in integration by parts is to choose u and dv to obtain a new integral that is easier to evaluate than the original. In general, there are no hard and fast rules for doing this; it is mainly a matter of experience that comes from lots of practice.
Guidelines
• There is a useful strategy that may help when choosing u and dv. When the integrand is a product of two functions from different categories in the following list , you should make u the function whose category occurs earlier in the list.
• Logarithmic, Inverse Trig, Algebraic, Trig, Exponential
• The acronym LIATE may help you remember the order.
Guidelines
• If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.
Guidelines
• If the new integral is harder that the original, you made the wrong choice. Look at what happens when we make different choices for u and dv in example 1.
xdxx cosxu cos
xdxdu sin
2
2xv
xdxdv
xdxx
xx
xdxx sin2
cos2
cos22
Guidelines
• Since the new integral is harder than the original, we made the wrong choice.
xdxx cosxu cos
xdxdu sin
2
2xv
xdxdv
xdxx
xx
xdxx sin2
cos2
cos22
Example 2• Use integration by parts to evaluate dxxex
Example 2• Use integration by parts to evaluate
xu dxedv x
dxxex
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
dxexedxxe xxx
Example 2• Use integration by parts to evaluate
xev
xu dxedv x
dxdu
dxxex
dxexedxxe xxx
Cexedxxe xxx
Example 3 (S):• Use integration by parts to evaluate xdxln
Example 3• Use integration by parts to evaluate
xu ln dxdv
xdxln
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
dxxxxdx lnln
Example 3• Use integration by parts to evaluate
xv
xu ln dxdv
dxx
du1
xdxln
dxxxxdx lnln
Cxxxxdx lnln
Example 4 (Repeated):• Use integration by parts to evaluate dxex x2
Example 4 (Repeated):• Use integration by parts to evaluate
2xu dxedv x dxex x2
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxedv x
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
dxexeexdxex xxxx 222
Example 4 (Repeated):• Use integration by parts to evaluate
xev
2xu dxedv x
xdxdu 2
dxex x2
dxxeexdxex xxx 222xu dxdu xev
dxedv x
dxexeexdxex xxxx 222
Cexeexdxex xxxx 2222
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
dxdv
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu xdxdu 2
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
u
duxxdxx2
1tan)(tan 1
1
0
1
Example 5:• Evaluate the following definite integral
xu 1tan
1
0
1 )(tan dxx
21
1
xdu
dxdv xv
21
1
0
1
1tan)(tan
x
xdxxxdxx
21 xu
dxx
du
2
xdxdu 2
u
duxxdxx2
1tan)(tan 1
1
0
1
10211
0
1 )1ln(2
1tan)(tan xxxdxx
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
10211
0
1 )1ln(2
1tan)(tan xxxdxx
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
10211
0
1 )1ln(2
1tan)(tan xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
)1ln(2
1tan)(tan 21
1
0
1 xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
002ln2
1
4)(tan
1
0
1 dxx
Example 5:• Evaluate the following definite integral
1
0
1 )(tan dxx
)1ln(2
1tan)(tan 21
1
0
1 xxxdxx
)01ln(2
10tan0)11ln(
2
11tan1)(tan 2121
1
0
1 dxx
2ln4
002ln2
1
4)(tan
1
0
1 dxx
Homework:Page 520
# 3-9 odd, 15, 25, 29, 31, 37